
\subsubsection{Mathematical model}
Objective:
\begin{equation}
\max{\prod_{i \in I}{u_i}}
\end{equation}

Constraints:
\begin{align}
&\forall_{i \in I}{x_i\ge 0} \\
&\forall_{i \in I}{c_i\ge 0} \\
&\forall_{i \in I}{y_i = x_i + c_i} \\
&\sum_{i \in I}{x_i} = G \\
&\forall_{i \in I}{u_i = c_i + \alpha_i \cdot G}
\end{align}

\subsubsection{AMPL model}

\begin{lstlisting}[language=c,style=outcode,caption=nash.model.ampl]
################# PARAMETERS
param N;                     # number of the agents
param G;                     # cost of the good
param alpha{1..10} >=0, <=1; # MRS
param y{1..10} >=0;          # endowment
################# VARIABLES
var x{1..N} >=0;             # contribution/tax
var u{1..N};                 # utility
var c{1..N} >=0;             # private constumption
################# CONSTRAINTS
subject to util {i in 1..N}:
	u[i] = c[i] + alpha[i] * G;
subject to provide_good:
	sum{i in 1..N} x[i] = G;
subject to budgets {i in 1..N}:
	y[i] = x[i] + c[i];
################# OBJECTIVE
maximize nash:
	prod{i in 1..N} u[i];
\end{lstlisting}

\subsubsection{Results}
\begin{lstlisting}[language=c,style=outcode,caption=Nash results for scenario A]
DONLP2: Success! KKT conditions satisfied
F = 3.871284695e+21
25 iters; 28 function, 21 gradient evals
1822 component constraint, 21 grad evals
final objective scaling = 0.0001
norm(grad(f)) = 8.55554e+19, Lagrangian violation = 6.89524e+14
feas. violation = 7.81597e-14, dual feas. violation = 0
###### NASH ######
\end{lstlisting}


\begin{table}[H]
\caption{Nash results for scenario A}
\begin{center}
\begin{tabular}{|r||r|r|r|r|r|}
\hline
$i$ & $\alpha_i$ & $y_i$ & $x_i$ & $c_i$ & $u_i$ \\
\hline 
1&0.55&91.15&0.00&91.15  &145.85\\
2&0.75&121.29&44.67&76.62&151.81\\
3&0.54&100.49&2.75&97.75 &151.81\\
4&0.23&93.48&0.00&93.48  &116.91\\
5&0.43&107.64&0.00&107.64&150.51\\
6&0.56&127.22&31.29&95.93&151.81\\
7&0.72&98.34&18.55&79.78 &151.81\\
8&0.67&74.08&0.00&74.08  &141.34\\
9&0.47&85.57&0.00&85.57  &132.36\\
10&0.54&100.62&2.75&97.87&151.81\\
\hline
\end{tabular}
\end{center}
\label{tab:nash_res_A}
\end{table}

